##
Dualizing Initial Algebras

Neil Ghani^{1}, Christoph Lüth^{2},
Federico de Marchi^{1}, John Power^{3}

^{1}Department of Mathematics and Computer Science,
University of Leicester

^{2}FB3 - Mathematik und Informatik, Universität Bremen

^{3}Laboratory for Foundations of Computer Science,
University of Edinburgh

Whilst the relationship between initial algebras and monads is
well-understood, the relationship between final coalgebras and
comonads is less well explored. This paper shows that the
problem is more subtle and that final coalgebras can just as easily
form monads as comonads and dually, that initial algebras form both
monads and comonads.

In developing these theories we strive to provide them with an
associated notion of syntax. In the case of initial algebras and
monads this corresponds to the standard notion of algebraic
theories consisting of signatures and equations: models of
such algebraic theories are precisely the algebras of the representing
monad. We attempt to emulate this result for the coalgebraic case by
defining a notion cosignature and coequation and then proving the
models of this syntax are precisely the coalgebras of the representing
comonad.

Christoph Lüth, 02.05.2002